Finite Group Databasebeta

← #13137749

Subgroups — #13137749

Order 1728.

394 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic, complemented
2#11721045C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720278C211yesmaximal, characteristic, complemented
8#10590512—31nomaximal
9#10588459—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×#6195—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4×C3⋊Dic9—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21#64101—31no—
22#63916—31no—
23C3×#809D311yescharacteristic
24C3×#807—31no—
25#63573—31no—
26#63563—31no—
27#63557—31no—
28#63500—31no—
29#63454—31no—
30#63454—61no—
31C9×#102—31no—
32#63392—31no—
33(C2×D9)⋊C8—31no—
34#63377—31no—
35#63366D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C22×D3×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C4×C3⋊D9—21no—
43C4×C3⋊D9—21no—
44C2×C3×Dic9—21no—
45C9×D12—21no—
46C9×D12—21no—
47#6195—42no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4885—91no—
51#4779—91no—
52C2×C4×C3⋊D3—31no—
53C2×C3×D12C2×D311yescharacteristic
54C2×#626—61no—
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—61no—
57C4⋊(C3⋊Dic3)—31no—
58C4×C3⋊Dic3—61no—
59C3×C4⋊Dic3—31no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62(C2×D3)⋊Dic3—61no—
63C2×D4×C9—31no—
64C2×#611—61no—
65C2×C4×D9—31no—
66C2×C4×D9—61no—
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C2×D311yescharacteristic
70#3676—61no—
71(C2×D9)⋊C4—61no—
72(C2×D9)⋊C4—62no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9C2×D311yescharacteristic
76C4×Dic9—61no—
77C4×Dic9—62no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C2×C3⋊D9—21no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C2×D3×C9—41no—
86C3×C4×C9—41no—
87C3×C4×C9#4111yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9—41no—
91C3×Dic9—41no—
92#938—91no—
93#895—91no—
94#892—91no—
95#892—91no—
96#892—181no—
97C3×#107—91no—
98C3×#102—31no—
99#832—91no—
100#821—91no—
101(C2×D3)⋊C8—91no—
102(C2×D3)⋊C8—91no—
103#817—91no—
104#809D911yescharacteristic, complemented
105#808—91no—
106#807—31no—
107C22×C3⋊D3—31no—
108C22×C3×D3—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—61no—
113C4×C3⋊D3—61no—
114C4×C3⋊D3—61no—
115C2×C3×Dic3—61no—
116C3×D12—21no—
117C3×D12—21no—
118#626—122no—
119C23×C9—61no—
120C22×D9—31no—
121C22×D9—61no—
122C3×C3⋊C8—21no—
123C3×C3⋊C8—61no—
124D4×C9—61no—
125D4×C9—61no—
126C2×C4×C9—22no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9C22×D311yescharacteristic
130#611—122no—
131C2×Dic9—21no—
132C2×Dic9—31no—
133C2×Dic9—61no—
134C2×Dic9—62no—
135C2×Dic9—61no—
136C4×D9—62no—
137C4×D9—62no—
138C4×D9—61no—
139C4×D9—61no—
140C8×C9—61no—
141C9⋊C8—21no—
142#482—271noSylow
143C2×C3×C9#12311yescharacteristic
144C2×C3×C9#13511yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—41no—
147D3×C9—81no—
148C2×C3×D4—31no—
149C2×#67—181no—
150C2×D12—181no—
151C2×D12C2×D911yescharacteristic
152C2×C4×D3—91no—
153C2×C4×D3—91no—
154C2×C4×D3—181no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4C2×D911yescharacteristic
160C3×C4⋊C4D3211yescharacteristic
161#217—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—182no—
166C4⋊Dic3—31no—
167C4⋊Dic3—91no—
168C4⋊Dic3—91no—
169C4⋊Dic3—181no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C4×Dic3—182no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8C2×D911yescharacteristic
175C22×C32D3×D411yescharacteristic
176C2×C3⋊D3—61no—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—61no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—41no—
182C32×C4—41no—
183C32×C4#23511yescharacteristic
184C32×C4#23711yescharacteristic
185C3⋊Dic3—61no—
186C3⋊Dic3—121no—
187C3×Dic3—121no—
188C22×C9—21no—
189C22×C9—121no—
190C22×C9—121no—
191C22×C9—121no—
192C22×C9D3×D411yescharacteristic
193C2×D9—62no—
194C2×D9—61no—
195C2×D9—61no—
196C2×D9—61no—
197C2×D9—121no—
198C2×D9—121no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—42no—
202C4×C9—41no—
203C4×C9#23511yescharacteristic
204C4×C9#23911yescharacteristic
205Dic9—41no—
206Dic9—62no—
207Dic9—61no—
208Dic9—122no—
209Dic9—121no—
210#121—271no—
211#117—271no—
212#107—91no—
213#106—271no—
214#102—31no—
215#102—271no—
216C22⋊C8—271no—
217C3×C9#48211yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C22×D3—21no—
220C22×D3—91no—
221C22×D3—91no—
222C22×D3—181no—
223C3×D4—61no—
224C3×D4—61no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—181no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232#67—362no—
233C2×Dic3—61no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—181no—
237C2×Dic3—182no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240D12—21no—
241D12—21no—
242D12—362no—
243C4×D3—182no—
244C4×D3—182no—
245C4×D3—181no—
246C4×D3—181no—
247C4×D3—181no—
248C4×D3—181no—
249C3×C8—61no—
250C3×C8—181no—
251C3⋊C8—21no—
252C3⋊C8—61no—
253C2×C32#88311yescharacteristic
254C2×C32#91111yescharacteristic
255C2×C32#91611yescharacteristic
256C3⋊D3—121no—
257C3×D3—81no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—241no—
262C2×C9#89411yescharacteristic
263C2×C9#91111yescharacteristic
264C2×C9#91911yescharacteristic
265D9—122no—
266D9—121no—
267C2×D4—31no—
268C2×D4—541no—
269C22×C4—271no—
270C2×C8—31no—
271C2×C8—91no—
272C4⋊C4—91no—
273C4⋊C4—271no—
274C4⋊C4D3×D911yescharacteristic
275C22⋊C4—541no—
276C22⋊C4—541no—
277C42—541no—
278C22×C3—21no—
279C22×C3—121no—
280C22×C3—121no—
281C22×C3—121no—
282C22×C3D4×D911yescharacteristic
283C22×C3#381111yescharacteristic
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—182no—
288C2×D3—181no—
289C2×D3—181no—
290C2×D3—181no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—181no—
294C2×D3—361no—
295C2×D3—361no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—361no—
302C3×C4#369711yescharacteristic
303C3×C4#369911yescharacteristic
304C3×C4#379611yescharacteristic
305C3×C4#379811yescharacteristic
306Dic3—121no—
307Dic3—182no—
308Dic3—181no—
309Dic3—181no—
310Dic3—362no—
311Dic3—361no—
312Dic3—361no—
313C32#477911yescharacteristic
314C9—21no—
315C9#488511yescharacteristic, complemented
316C23—61no—
317C23—271no—
318D4—61no—
319D4—61no—
320D4—1082no—
321C2×C4—21no—
322C2×C4—181no—
323C2×C4—271no—
324C2×C4—541no—
325C2×C4—541no—
326C2×C4—541no—
327C2×C4C2×D3×D911yescharacteristic
328C8—61no—
329C8—181no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—241no—
334C2×C3#6344511yescharacteristic
335C2×C3#6347311yescharacteristic
336C2×C3#6347811yescharacteristic
337C2×C3#6385311yescharacteristic
338C2×C3#6392711yescharacteristic
339C2×C3#6395011yescharacteristic
340D3—81no—
341D3—362no—
342D3—361no—
343D3—361no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—541no—
348C22—541no—
349C22—541no—
350C22#8866811yescharacteristic
351C4—41no—
352C4—361no—
353C4—541no—
354C4—1081no—
355C4#8863911yescharacteristic
356C4#8864011yescharacteristic
357C3—21no—
358C3#1058845911yescharacteristic, complemented
359C3#1059051211yescharacteristic
360C2—241no—
361C2—1081no—
362C2#1172095711yescharacteristic
363C2#1172117511yescharacteristic
364C2#1172121611yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721045C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#482
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721748C2semidirect
3. C9×#809C2semidirect
4. C3×#63366C2semidirect
5. #11720295C2semidirect
6. #11720282C2semidirect
7. #11720278C2semidirect
35. #63366D3semidirect
104. #809D9semidirect
217. C3×C9#482semidirect
315. C9#4885semidirect
358. C3#10588459semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117217482
#117210452
C9×#8092
C3×#633662
#117202952
#117202822
#117202782
#105905123
#105884593